Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), and let $n,m\in\mathbb N$. Let $X\subseteq \mathbb A_k^n$ and $Y\subseteq \mathbb A_k^m$ be affine varieties. Define their coordinate rings by
Then the assignment sending a regular map $\varphi:X\to Y$ to the pullback homomorphism $\varphi^*:k[Y]\to k[X]$, defined by $\varphi^*(g)=g\circ\varphi$ after identifying coordinate ring elements with regular functions, gives a bijection between regular maps $X\to Y$ and unital $k$-algebra homomorphisms $k[Y]\to k[X]$.